Most English speakers can parse the sentence just fine when they hear it spoken.

Not so. I've tried this on a few dozen students (post-secondary, many of them post-baccalaureate, perhaps 50% ESL) over the years; of those who have not heard it previously, all others say "what? that's not a sentence". I don't recall a single counter-example.

That's surprising to hear. I don't think I've ever found someone that understood it without an explanation. I like using it as an example of how confusing English can be to parse.

English is tough, though through thorough thought it can be understood.

Shouldn't that be "English is tough, though through thorough thought it can be taught"?

I would not have been able to tell which buffalos were verbs, which were nouns and which were proper nouns without the rephrasing of the sentence on Wikipedia. In fact even with that rephrasing I still find it difficult to tell which buffalo in the original sentence is what.

the argument is nevertheless valid: the parser does not process any additional audible signal